Moderate deviations for the self-normalized random walk in random\n scenery

2020/01/16 by Tal Peretz, Peretz, Tal
Decision Sciences · Mathematics · #60F10 (Primary) 60G50 #60K37 (Secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2001.05736

openalex publication_date 2020/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be an infinite connected graph with vertex set V. Let S_n: n \∈\n mathbb N₀ be the simple random walk on G and let \ξ(v) : v \∈ V\n be a collection of i.i.d. random variables which are independent of the\nrandom walk. Define the random walk in random scenery as T_n = \∑_k=0ⁿ\n\ξ(S_k), and the normalization variables V_n = (\∑_k=0ⁿ\n\ξ²(S_k))^1/2 and L_n,2 = (\∑_{v \∈ V} \ℓ²_n(v))^1/2. For G=\n mathbb Z^d and G = mathbb T_d, the d-ary tree, we provide large\ndeviations results for the self-normalized process T_n \√(n)/(L_n,2V_n)\nunder only finite moment assumptions on the scenery.\n

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